Questions tagged [groupoids]

A groupoid is a small category in which every morphism is an isomorphism. They arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag. Also, use other tags such as monoid or category-theory, if needed.

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Categorical Intuition of Path Induction

I am trying to understand path induction from the trinitarian point of view. So far I understand the informal intuition of path induction from a homotopical and computational point of view. But I am ...
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Continuity of Norm of Regular Representations on Étale Groupoids

Let $\mathcal G$ be a locally compact, Hausdorff, étale groupoid. Following the notation found in Sims' notes (see here), for $x\in\mathcal G^{(0)}$, let $\lambda_x:C_c(\mathcal G)\to\mathbb B(\ell^2(...
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Is the Mathieu Groupoid $M_{13}$ even special, besides its construction?

The Mathieu groupoid $M_{13}$ has a beautiful construction from a "sliding block puzzle" produced from the projective plane of order $3$; put $12$ labelled counters on $12$ of the $13$ ...
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A groupoid that is not a fundamental groupoid

There is a nice result that says for any group $G$, there exists a topological space $X$ such that the fundamental group $\pi_{1}(X)$ of $X$ is isomorphic to $G$. The proof is not terribly complex: we ...
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Universal cover of a connected groupoid

A group $G$ is a category with exactly one object and all invertible morphisms, which are tue usual elements of the set theoretic group which may be denoted by $|G|$. Now, for a $G$-set $S$, $\textit{...
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1-dimensional foliation of surfaces with prescribed graph of foliation

Definition of the graph of a foliation Let we have a $k$ dimensional foliation of an $n$ dimensional manifol $M$. One associates to this foliated manifold a (not necessarily Hausdorff) ...
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Isotropy subgroup in groupoid

Let $G$ be a groupoid. I am struggling to prove some facts from the definition directly. $(x,y) \in G^{(2)}$ iff $d(x)=r(y)$ where $G^{(2)}$ is the set of composible pairs. Let $G^0$ be the unit space....
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Without totality: Semigroupoid, Small Category, and Groupoid - how not be closed possible?

Semigroupoid, Small Category, and Groupoid have group like structures, see https://en.wikipedia.org/wiki/Monoid#Relation_to_category_theory But there Semigroupoid, Small Category, and Groupoid do not ...
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Groupoid structure of a kernel pair

I would have probably very short and simple question. In https://ncatlab.org/nlab/show/%C4%8Cech+nerve there is a statement Čech nerve of a morphism $f$ is the internal nerve of the internal groupoid ...
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Functors making functions natural transformations and vice-versa.

I apologize in advance if this is naive. In this answer Conjugation in a groupoid it is said that given a groupoid $\mathcal G$, and an arbitrary function $\mu:\mathcal G_0\to \bigcup_{x\in \mathcal ...
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(non-)Existence of an endofunctor of $F$ on $g/{\bf B}G^\circlearrowleft$ given the definition of $F$ on objects.

In few words: given an element of a group $g\in G$ given an I'd like to define and endofunctor $$F:g/{\bf B}G^\circlearrowleft\to g/{\bf B}G^\circlearrowleft$$ I have defined $F$ on objects and I'd ...
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What are groupoids?

As far as I know a groupoid is a category in which every morphism is invertible, however I do not understand this alternative definition: Let $\mathcal{C}$ be a category with coproducts, $\mathcal{T}$ ...
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What is the practical benefit of a groupoid structure?

Pardon if this question is vague or misguided, I'm a CS person who only dabbles in math. In a groupoid all morphisms are isomorphisms. So then, any two objects with a morphism between them must be ...
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Is a connected groupoid uniquely determined by its objects and fundamental group?

A small connected groupoid is such a category $G$ that: $\text{Ob}(G)$ is a nonempty set $\text{Hom}_G(a,b)$ is a nonempty set for all $a,b\in\text{Ob}(G)$ $f$ is invertible for each $f\in\text{Hom}...
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Classical Galois Theory and Groupoids?

Given your favorite field $k$, we can build its groupoid of algebraic closures in the obvious way: Look at the category of all possible algebraic closures with field isomorphisms as the arrows. This ...
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Good book for self-study of Magmas/Semigroups/etc.?

I'm currently an undergrad in my second semester of Abstract Algebra. We've covered groups, rings, fields, all that fun stuff. I'm working with Shahriari's "Algebra in Action" as well as ...
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Explicit description of pullback of $(2,1)$-categories.

Let us consider the ordinary category of $(2,1)$-categories. Its objects are groupoid enriched categories, and its morphisms are 2-functors. Is there an explicit way to define the objects, morphisms ...
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Pushout of groupoid

I'm learning category theory. There's a homework question asking for a pushout of groupoid. Suppose $C_0,C_1,C_2$ are groupoids and denote $f_i:C_0\to C_i\quad i=1,2$ the functor. I have managed to ...
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Hopf "algebroid" structure of a groupoid convolution algebra?

To male thinks simple as possible, lets say we have a discrete group $G.$ Then the then the group algebra $\mathbb{C}[G]$ (of finitely supported complex valued functions on $G$) has a convolution and ...
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adjunctions from either the category of small categories or the category of groupoids

I'm looking for examples of adjunctions involving either the category Gpd of groupoids or the category Cat of small categories. The insertion of Gpd into Cat has both a left and a right adjoint, so ...
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Equivalence Between Category of Covering Spaces and Category of Sets with an Action By The Fundamental Groupoid

Let $\Pi_1(X)$ be the fundamental groupoid of a locally path-connected topological space $X$ and define $\Pi_1(X)-\mathbf{Sets}$ to be the category of sets equipped with an action by the fundamental ...
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Functor From Category of Covering Spaces to Category of Sets Equipped With An Action By The Fundamental Groupoid

I have some problems with the understanding of the connection between covering spaces and the fundamental groupoid. Let $X$ be a topological space and let $\Pi_1(X)$ denote the fundamental groupoid of ...
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Proving that a groupoid is a group, knowing the following properties

(G, ·) is a groupoid. Prove that if it has the following properties it is also a group: $$1) (a · b) · c = a · (b · c), (\forall)a, b, c \in G;$$ $$2) (∃)u ∈ G : u · a = a · u = a, (∀)a ∈ G;$$ $$3) (∀)...
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The small concrete category $C = (\mathbb{Z_2}, +)$?

I'm grappling with the question of viewing a group as a category, which as I understand it means that if $(G,+)$ is the group in question, then the group can also be thought of as a small concrete ...
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What's the Internal language of a groupoid?

Is there any existing literature on what the internal language of a groupoid might look like? Please excuse the syntax bashing of this amateur, but i came up with: symmetry (structural rule): $$ \frac{...
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classifying space induces a equivalence of categories between PBun$_G(M)$ and $\Pi(M,BG)$ for finite groups $G$

Let $G$ be a finite group, $BG$ its classifying space and M a manifold. Then it is mentioned in https://arxiv.org/abs/1705.05171 (Remark 2.3 d) that there is an equivalence of categories $$ \Pi (M,BG) ...
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Notation for the scope of definition of a partial binary operation of a groupoid

I have a groupoid ${\displaystyle (G,\ast )}$ with a partial binary operation ${\displaystyle *:G\times G\rightharpoonup G}$. For every $(a,b)\displaystyle ∈G\times G$, $\displaystyle *$ is defined if ...
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Name for semigroupoid-like structure applicable to a water flow network in graph theory

Hope you are well. I am studying flow networks (flows of water) in graph theory. By flow network I mean a graph $G = (V,E)$ where $V$ is a set of $ℝ_{>0}$ labelled vertices (water tanks filled with ...
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Group isomorphism between identity morphisms of fundamental groupoid and fundamental group of topological space

For any topological space $X$, I have defined $C=\Pi_1(X)$ by objects $\text{Ob}(C)=X$ and morphisms $\text{Hom}_C(x,y)=\text{HPath}(x,y)$, where $\text{HPath}(x,y)$ denotes the set of homotopy ...
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2 votes
2 answers
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Homotopy cardinality of weak quotients

Let $G$ be a group (regarded as a one-object groupoid) acting on a groupoid $X$ (i.e.: a functor $F: G \to \mathsf{Groupoids}$ sending the unique object of $G$ to $X$). Denote with $X//G$ the “weak ...
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Definition of Haar systems, especially for groupoids $C^*$-algebra

In A Groupoid approach to $C^*$-algebras Jean Renault introduces left Haar systems. Say $G$ is a groupoid. Let $\Lambda=\{\mu^u,\,\,:\,\,u\in G^0\}$ be a family of Haar measures on $G$. Renault ...
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Help with understanding the definition of a Hodge groupoid

I am reading this PhD thesis and I can't understand definition 6.21: (Hodge groupoid). $(X/S)_{Hod}$ is a groupoid whose object object is $X\times \Bbb A^1_S$ and whose morphism object $N_{Hod}$ is ...
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How units of a Lie groupoid act on a manifold?

The definition of a Lie groupoid action given in Sébastien Racanière's notes (up to notation) says that the action of a Lie groupoid $\mathcal{G} \rightrightarrows M$ on a smooth manifold $Q$ consists ...
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Cancellation law in a Lie groupoid

I'm playing around with the definition of a Lie groupoid following Eckhard Meinrenken's notes. I read the thing for the first time in my life like an hour ago, so assume that I don't know anything ...
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4 votes
3 answers
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Simplicial sets which are not Kan complexes

A Kan complex is a simplicial set satisfying the horn-filler condition. What examples are there of simplicial sets which do not satisfy the horn-filler condition? In particular, what simplicial sets ...
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Every normal subgroupoid is the kernel of a groupoid morphism

Sorry for the block of text from Kirill Mackenzie's chapter on groupoids. I have proven that $\natural,\natural_0$ is a groupoid morphism and that this quotient construction satisfies the groupoid ...
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Why Isn't a piecewise-injective groupoid morphism injective?

The following is a question I had while reading Kirill Mackenzie's first book (RIP) on groupoids. He says that if a groupoid morphism is base-injective and piecewise-injective, then it is injective. I ...
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Exact sequences of Groupoids?

My question is basically: Is there a good notion of complexes of groupoids and there exactness? I haven't found a definition online.
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Bijection Between Maps of Covering Spaces and Maps of Fundamental Groupoid Covers

I'm reading May's Concise Course in Algebraic Topology. He states: My question is about this last corollary. How does the bijection $\text{Cov}(E, E') \to \text{Cov}(\Pi(E), \Pi(E'))$ immediately ...
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4 votes
1 answer
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Quotient by a topological groupoid.

Let $G$ be a group acting on a topological space $X$, then the quotient map $X \to X/G$ is open. I want to ask, whether this fact generalizes to orbit spaces of groupoids. More precisely: Let $G$ be a ...
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$G$-set are groupoids of a fibred category

Take the category of all G-sets for different groups $G$. Each $G$-set is the groupoid $G \times \Omega \to \Omega$ (the first projection is the source and the action map is the target). I am not ...
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3 votes
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Why can't the 3-type of $S^2$ be delooped?

In 2.2 of this paper, the argument runs through the fact that the 3-type of $S^2$ cannot be delooped. Why not? I understand that this is probably a fairly basic fact of homotopy theory (hence neither ...
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Number of fixed point free permutations by groupoid cardinals

It is well known that the number $D_n$ of derangements (fixed point free permutations) on a set of $n$ elements is exactly $\left[\dfrac{n!}e\right]$, the closest integer number to $\dfrac{n!}e$. ...
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9 votes
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$\pi_1(S^1, 1)$ via the fundamental groupoid

I'm currently reading Ronnie Brown's Topology and Groupoids and am stuck on a small detail of his computation of the fundamental group of the circle (in particular his computation of the group's ...
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2 votes
1 answer
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Functor from $BG \to \text{Gpd}$ induces functors for each $g \in G$.

I'm reading Groupoids and Stuff. Page 7. Definition 1.3.1. A group action of a a finite group $G$ on a groupoid $X$ is a functor $A : B G \to \text{Gpd}$ such that $A(I) = X$ where $I$ is the unique ...
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Groupoids all of whose subcategories are themselves groupoids

It is known that every submonoid of a group $G$ is a subgroup if and only if $G$ is a periodic group, i.e. all of its elements have finite order. The following question is a generalization of the ...
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How to realize the map $\eta$ globally?

I have a given map $\Phi: \mathcal{G}\longrightarrow \mathcal{H}$ between two groupoids such that $\Phi_g: \mathcal{G}_x\longrightarrow \mathcal{G}_y$ is a functor between the groupoids $\mathcal{G}_x$...
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7 votes
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Riehl's Category Theory in Context - Exercise 1.5.vii without Axiom of Choice

From Emily Riehl, Category theory in context: Exercise 1.5.vii. Let $\mathbf{\mathsf G}$ be a connected groupoid and let $G$ be the group of automorphisms at any of its objects. The inclusion $\...
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Exemples of applications of "groupoidification" to linear algebra

I just read Baez's very nice blog notes about groupoidification, and around the beginning, he states : "From all this, you should begin to vaguely see that starting from any sort of incidence ...
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Is the 2-Category of Groupoids Locally Presentable?

I am wondering if the 2-Category of groupoids is Locally Presentable? Locally presentable means the category is accessible and co-complete. Edit: It has been pointed out that the category of ...
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